Maximal and pointwise ergodic theorems for word-hyperbolic groups

Abstract

Let $\Gamma$ denote a word-hyperbolic group, and let $S=S^{-1}$ denote a finite symmetric set of generators. Let $S_n = \{w: |w| = n \}$ denote the sphere of radius $n$, where $|\cdot|$ denotes the word length on $\Gamma$ induced by $S$. Define $\s_n \deq (1/{\#S_n}) \sum_{w \in S_n}w$, and $\mu_n=(1/(n+1))\sum_{k=0}^n \s_k$. Let $(X,{\cal B},m)$ be a probability space on which $\Gamma$ acts ergodically by measure-preserving transformations. We prove a strong maximal inequality in $L^2$ for the maximal operator $f_{\mu}^*=\sup_{n\ge 0}|\mu_n f(x)|$. The maximal inequality is applied to prove a pointwise ergodic theorem in $L^2$ for exponentially mixing actions of $\Gamma$ of the following form: $\mu_n f(x) \to \int_X f\,dm$ almost everywhere and in the $L^2$-norm, for every $f\in L^2(X)$. As a corollary, for a uniform lattice $\Gamma \subset G$, where $G$ is a simple Lie group of real rank one, we obtain a pointwise ergodic theorem for the action of $\Gamma$ on an arbitrary ergodic $G$-space. In particular, this result holds when $X=G/\Lambda$ is a compact homogeneous space, and yields an equidistribution result for sets of lattice points of the form $\Gamma g$, for almost every $g\in G$.